„There was a time when newspapers said that only

twelve men understood the theory of relativity.

I do not believe that there ever was such a time...

On the other hand,

A very elementary approach to Quantum mechanics

R.P. Feynman The Character of Physical Law (1967)

I think it is safe to say that

no one understands quantum mechanics“

let´s approach some aspects of qm anyway

Experimental facts:

Light has wave (interference) and particle properties

Plot from

maxkinE

Existence of photons

E

Energy of the quantumPlanck’s const.

Frequency

Radiation modes in ahot cavity providea test of quantum theory

Energy of a free particle

2E mc 22 2 40c p m c where

0

21 /

mm

v c

Consider photons

0 0m v cand E with

cp h h

pc c

c ck

or p k

; p mv

Dispersion relation for light

Electrons (particles) have wave properties

Today: LowEenergyElectronDiffraction standard method in surface science

Figures from

p k applicable for “particles” de Broglie

LEED

Fe0.5Zn0.5F2(110)

232 eV

top view

(110)-surface

Implications of the experimental facts

Electrons described by waves:( )( , ) i kx tx t A e

Wave function (complex for charged particles like electrons)

2*( , ) ( , ) ( , ) ( , )x t dx x t x t dx x t dx Probability to find electron at (x,t)

Which equation describes the temporal evolution of ( , )x t

Schroedinger equation

Erwin Schroedinger

Can’t be derived, but can be made plausible

Let’s start from the wave nature of, e.g., an electron:

( )( , ) i kx tx t A e and take advantage of p k ;E

( ) /( , ) i px Etx t A e x

( ) /( , ) i px Etipx t A e

x

( , )

ipx t

i

( , ) ( , )i x t p x tx

p i px

In complete analogy we find the representation of E

t

( ) /( , ) i px Etx t A e

( ) /( , ) i px EtiEx t A e

t

( , )

iEx t

i

( , ) ( , )i x t E x tt

E i H

t

Schroedinger equation for 1 free particle

Hamilton function of classical mechanics2

2

pH

m ; H=E total energy

of the particle

p i px

E i Ht

2 2

2( , ) ( , )

2x t i x t

m x t

1-dimensional

In 3 dimensions , ,p i i px y z

2

( , ) ( , )2

r t i r tm t

where 2

Schroedinger equation for a particle in a potential

Classical Hamilton function:2

( )2

pH V r

m

2

( ) ( , ) ( , )2

V r r t i r tm t

H

Hamilton operator

( , ) ( , )H r t i r tt

Time dependentSchroedinger equation

If H

independent of time like 2

( )2

H V rm

only stationary Schroedinger equation has to be solved Proof:

Ansatz: ( , ) ( )iE t

r t r e

( , ) ( , )H r t i r tt

(Trial function)

( ) ( )i iE t E t

H r e i r et

( ) ( )H r E r

Stationary Schroedinger equation

Solving the Schroedinger equation (Eigenvalue problem)

Solution requires: -Normalization of the wave function according2 3( ) 1r d r

Physical meaning: probability to find the particle somewhere in the universe is 1

-Boundary conditions of the solution:

and have to be continuous when merging piecewise solutions

Note: boundary conditions give rise to the quantization

Particle in a box:

x

2( ) sinn

nxx

L L

Eigenfunctions

2 2 2

21,2,3,...

2n

nE n

mL

Eigenenergies

Quantum numberDetails see homework

Heisenberg‘s uncertainty principle

It all comes down to the wave nature of particles

( ) /( , ) i px Etx t A e

Wave function given by a single wavelength h

p

Momentum p precisely known, but where is the particle position

-P precisely given

-x completely unknown

Wave package

Particle somewhere

in the region x

Fourier-analysis

11

hp

22

hp

Particle position known with uncertainty x

Particle momentum known with uncertainty p2

x p

Fourier-theorem

In analogy2

E t